Design and Optimization of Flexible Utility Systems Subject to Variable Conditions

DESIGN AND OPTIMIZATION OF FLEXIBLEUTILITY SYSTEMS SUBJECT TO VARIABLECONDITIONSPart 2: Methodology and Applications

O. Aguilar, S. J. Perry, J.-K. Kim� and R. Smith

Centre for Process Integration, School of Chemical Engineering and Analytical Science, The

University of Manchester, Manchester, UK.

Abstract: After having described the development of new models for the energy equipment usedin industrial utility systems in the first part of this paper, this second part explains how such a mod-elling framework has been developed into an integrated methodology for the design and optimiz-ation of utility plants that exploits the flexibility of these systems. The different types of problems tobe addressed (i.e., grassroots design, retrofit and operational) are identified in terms of their majorissues and tradeoffs, giving emphasis to the fact that these tasks require a robust optimization pro-cedure to handle industrial cases. Mechanical driver selection is incorporated in the proposedstrategy. Several examples demonstrate the applicability and the potential of the suggestedapproach to provide significant economic benefits when applied to industrial problems.

Keywords: energy systems; design; operation; cogeneration; flexibility.

INTRODUCTION

The design and operation of industrial utilitysystems offer multiple degrees of freedom(e.g., equipment sizes, number of units, andtheir loads) that can be exploited to reducecapital and/or operating costs. However,minimizing such expenditure also representsa challenging task, not only due to thehighly-combinatorial computations involved,but also because of the strong interactionsbetween the equipment. This means thatchanging any variable of the problem (e.g.,for a single unit) could potentially have animpact on the rest of the plant. Therefore,the whole utility system must be simulated inorder to take into account all the units anddetermine the plant-wide consequences ofany proposed modification. Moreover, only inthis way is it possible to assess whichdesign and/or operational options actuallyimprove the overall economics of the utilitysystem. For instance, increasing the size orload of a steam turbine to boost its power pro-duction and reduce operating costs (i.e., elec-tricity import) would be effective only ifrelatively inexpensive steam can be producedwithout compromising the operation of otherequipment, or incurring higher penalties fromchanges in the rest of the units.On the other hand, even if simulation tools

are readily available, there is still the difficulty

of evaluating the large number of optionswithin the solution space of a given problem.However, once a mathematical model of theutility plant has been constructed (i.e., by link-ing the individual models of all the items withinthe system), it is possible to employ compu-tational programming routines to comparequickly and effectively such alternatives.Hence, the design and operation of a utility

plant can be posed as mathematical optimiz-ation tasks to systematically search for theoptions that minimize/maximize a givenobjective (e.g. to reduce costs), while meetingthe specified demands and other problemconstraints.Additionally, in order to solve such tasks

taking into account variable operating con-ditions, several scenarios or operationalperiods might be defined. One of widely-used techniques to enhance the flexibility ofchemical processes is a multiperiod optimiz-ation (Varvarezos et al., 1992), where differentlengths for period can be defined to reflectvariations of certain operational parameters.Therefore, in the present work, a multiperiodapproach is applied, which is able to modifythe relative length and number of scenariosto better fit the expected operational fluctu-ations over a certain time horizon (e.g.,hours, days, months or years).Also, in order to avoid large economic penal-

ties, it is necessary to fully exploit the flexibility

1149 Vol 85 (A8) 1149–1168

�Correspondence to:Dr J.-K. Kim, Centre forProcess integration, Schoolof Chemical Engineering andAnalytical Science, TheUniversity of Manchester,P.O. Box 88, ManchesterM60 1QD, UK.E-mail: [email protected]

DOI: 10.1205/cherd06063

0263–8762/07/$30.00þ 0.00

Chemical EngineeringResearch and Design

Trans IChemE,Part A, August 2007

# 2007 Institutionof Chemical Engineers

of the utility system by considering that the all units can berun at partial load. Note that these considerations make thesolution procedure more complex because, not only equip-ment sizes must be established, but also their correspondingoperational conditions (e.g., on/off status and load) in eachscenario. Nevertheless, the proposed (linear) formulationhas been implemented into a robust (MILP) optimizationthat guarantees a global optimum solution, and can handleproblems of the size and complexity commonly found inindustry.Using non-linear models or expressions for unit operations

can be considered in the optimization to improve the accu-racy for estimating performance or characteristics of unitoperations at the expense of computational efforts. Thiscan be an initiative to employ non-linear models in the optim-ization if better accuracy from the non-linear models can beensured. However, it should be noted that parameters forunit models (shown in Part 1 of this paper) in this study areobtained from the regression of performance data or charac-teristics. A certain degree of errors are also existed althoughnon-linear expression is used for unit models, because modelparameters for non-linear expression are based on theregression as well (see Figure 5 of Part 1 of this paper).Because of this, there is no significant difference in the pro-posed study, in terms of unit model accuracy, betweenusing linear and using non-linear models.The first step to apply the suggested approach is to specify

the aim of the study (i.e., its objective function). It is thennecessary to define the input data and constraints to properlycharacterize a particular problem. The solution method mustdetermine the optimum value of all the variables that rep-resent the best way (as indicated by the objective of theanalysis) of designing and/or operating the plant. Once amultiperiod problem has been defined, the correspondingoptimization can be performed to determine the final designof the system, together with its operation policy in every scen-ario to minimize, for example, overall capital and/or operatingcosts. It is important to note that all the periods must be con-sidered at the same time in order to obtain a single plant con-figuration to reconcile the varying conditions (i.e., otherwisedifferent designs would result from each scenario).When variable operating conditions are selected or deter-

mined in this study, the value is chosen to reflect operatingconditions in a collective way (e.g., average value) within aperiod of interest. If deviation or uncertainty during the par-ticular period is large, a period with small time interval istaken to maintain the confidence in choosing appropriateoperating conditions. On the other hand, model uncertaintyin the optimization problem can be formulated such that devi-ation from nominal value is reflected, for example, by introdu-cing flexibility index or analysis (Swaney and Grossmann,1985a, b; Varvarezos et al., 1995). However, this needs cer-tain degree of user’s engineering decisions (e.g., how toselect nominal or average value, how to measure or deter-mine variation of variables or parameters). Therefore, the cur-rent study aims to develop a generic optimization frameworkfor utility systems under uncertainty, rather than exploringhow to define model uncertainty existed in practice.

TYPES OF PROBLEMS

Depending on the specified objective function, data, vari-ables to be solved, and their constraints, problems around

utility systems can be divided into three main groups: oper-ational, retrofit and grassroots design. A major feature of thesuggested methodology is that these three types of taskscan be addressed with a common optimization framework,considering design and operational variables simultaneously.

Operational Problems

These problems apply to existing utility systems in whichstructural modifications are not contemplated. Thus it isnecessary to establish the operational conditions throughoutseveral scenarios that minimize/maximize a given taskobjective (e.g., reduce overall costs or increase power pro-duction), while meeting process demands and other con-straints. Note that normally it is required to evaluate variousoperational periods in order to determine how to cope cost-effectively with different circumstances, such as plant start-up, shut-down, emergency cases and demands variations.In order to define a particular operational problem it isnecessary (apart from specifying an aim for the task) thefollowing data:

(1) Operational scenarios: number and duration of each timeperiod to be considered in the problem.

(2) Site conditions: this includes information such as ambienttemperature and altitude, along with energy and waterprices applicable to each operating scenario.

(3) Process demands: energy/water requirements fromexternal users in each period (e.g., electricity, mechanicalpower, steam, cooling water).

(4) Configuration and performance of the existing plant:types, number, sizes and interconnections of existingequipment units together with their part-load performancedata (i.e., to fit the models, if necessary).

This information is required not only for the general settingof the task (i.e., scenarios, conditions and demands), but alsoto build a mathematical model of the existing system. Thedata is employed to define a set of constraints that willapply to a particular problem, which correspond to the follow-ing physical and practical restrictions:

(1) Satisfying process requirements through all scenarios.(2) Meeting mass and energy balances.(3) Equipment performance: according to the mathematical

models derived previously, there is a specific relationbetween the input and output of each unit, which alsodepends on its size and load.

(4) Equipment operational limitations: equipment cannotoperate above its maximum limit (i.e., size) and oftenexhibit a minimum operational load.

Even after the constraints of a specific task have beenestablished there are still many ways in which the utilitysystem might satisfy them due to the large number of inter-connected units typically found in industrial applications.Consequently, these degrees of freedom represent an exten-sive solution space from which the decisions that minimize/maximize the objective of the problem might be obtained.The specific variables that must be defined and/or calculatedin an operational task are:

(1) On/off equipment status (i.e., whether a unit is running ornot in each operational scenario).

(2) Equipment loads (i.e., actual input and output of eachpiece of equipment in each operational scenario).

Trans IChemE, Part A, Chemical Engineering Research and Design, 2007, 85(A8): 1149–1168

1150 AGUILAR et al.

(3) Other operating conditions: pressure, temperature andflowrate for all steam/water streams, fuel consumption,power production, and so on in each operating period.

(4) Overall plant parameters: the operating variables that areweighed and added for all the scenarios (e.g., overallcosts, fuel consumption, emissions).

(5) Overall objective of the analysis: single or several (over-all) plant parameters might be considered as the taskobjective to be minimized/maximized.

As mentioned previously, the number of variables is nor-mally very large and interrelated. In principle, many of the vari-ables might affect the entire system. In order to illustrate thecomplexity of operational problems consider the small flow-sheet of Figure 1, for which it is required to find the mostcost-effective way of supplying medium-pressure (MP)steam. As shown in the diagram, there are several pathsthrough which this utility can be delivered to processes andeach has different system-wide economic implications. Forinstance, increasing the load of a boiler would improve its effi-ciency (i.e., decrease the fuel required per kilogram of steamproduced), but the loads of other units would be also reducedwith a corresponding increase in their specific fuel consump-tion, which can have different economic consequencesdepending on the fuels or fuel mixtures to be consumed inthese units. On the other hand, the effects on power productionmust be also taken into account, since reducing the steam fromheat recovery units (i.e., HRSG) would usually involve adecrease in GT electric output. Moreover, different loads onsteam turbines will affect not only power generation but alsosteam header temperature, which might induce a rise in let-down flow and further fuel consumption inside steam genera-tors. As can be appreciated, a large number of complexcalculations are required to address operational problemseven for simple cases. Furthermore, the alternatives that mini-mize/maximize the task objective can only be determined in aplant-wide context simulating and optimising the entire system.

Retrofit Design Problems

In retrofit design problems there is scope for making struc-tural modifications (i.e., some investment capital is available).Hence, it is required to determine the types and sizes of newunits (if any) to be installed, their number and connectionswithin the current system, and the operational conditions of

the whole plant (new and old equipment throughout all scen-arios). This will correspond with minimizing/maximizing thetask objective (e.g., reduce operating and/or capital costs),while meeting the corresponding constraints. In fact, theseproblems are a direct extension of pure operational onesand, consequently, they need the same information as theformer to set up the task and make the mathematical modelof the existing plant. However, retrofit cases need the follow-ing additional data related to new design options:

(1) Investment available (i.e., the maximum amount ofmoney available to install new equipment).

(2) Extended configuration alternatives: potential new unitsmust be included in a superstructure of configurationchoices.

(3) Capital cost functions.(4) Annualization factor: if total cost is considered as the task

objective, it is necessary to define the relative weight ofcapital costs compared with operational ones.

As discussed previously, the data also establishes a set ofconstraints that define a particular task. In fact, the types ofrestrictions for pure operational problems are also applicableto retrofit cases and it is only necessary to include a fewadditional constraints dealing with new design alternatives:

(1) Maximum investment available: the initial cost of newequipment cannot go beyond this limit.

(2) Size limits: each type of new unit being selected must bewithin practical equipment sizes.

(3) Maximum number of new units: new pieces of equipmentcannot go beyond the alternatives provided in the super-structure of extended configuration alternatives.

All the operating degrees of freedom valid for pure oper-ational problems also apply to retrofit cases.However, there are supplementary constraints to establish

the (new) configuration of the utility plant. Such structuraldegrees of freedom correspond to basic unknowns in thedesign, namely:

(1) Types, number and connections for new equipment units(i.e., which units should be selected from the superstruc-ture of extended alternatives).

(2) Sizes of new units: apart from selecting specific units fromthe superstructure it is necessary to size each of them.

(3) Existing units to be removed or replaced (i.e., if someequipment is not used at all it can be removed).

As with operational problems, typical industrial utility sys-tems can satisfy the constraints in many different ways. Forexample, Figure 2 shows several new-equipment alternatives(i.e., extended superstructure) for an existing site, which rep-resent in the suggested approach a continuous range (i.e.,an infinite number) of options, given that each of the potentialnew units can be of any size within practical limits. Further-more, since all the operational issues apply not only to theexisting elements, but to the new ones also, all design andoperational variables are also strongly interrelated. Forinstance, a larger boiler is more efficient and cheaper thantwo halfsized ones, but it would offer less operational flexibility.Consequently, in comparison with pure operational problems,retrofit cases involve an even greater number of variables,constraints and calculations that also require the simulationof the whole system in order to compare the overall economicconsequences of both design and operating decisions.

Figure 1. Flowsheet section illustrating some possible paths to supplyMP steam.


DESIGN AND OPTIMIZATION OF FLEXIBLE UTILITY SYSTEMS 1151

Grassroots Design Problems

This type of problem applies when it is necessary toestablish the design of a whole new utility system togetherwith its operational conditions (throughout several scenarios)that minimize/maximize the task objective (e.g., reducecapital plus operating costs), while satisfying the corre-sponding constraints. In terms of data requirements, grass-roots cases do not need the specification of any existingequipment. However, it is still necessary to supply thesame operating information as with previous types of pro-blems. In comparison with retrofit tasks, a larger superstruc-ture of alternatives for new equipment should normally beprovided.Although data requirements for grassroots problems is

(nominally) less than for retrofit cases, the number of appli-cable constraints and variables are, in general, higher dueto the increased options of new equipment for the wholeplant. Moreover, the solution space from which the variablesminimizing/maximizing the task objective might be obtainedis much larger, as every unit in the configuration superstruc-ture can also be of any size within practical limits. In thissense, pure operational and retrofit problems are particularcases of a more generic grassroots design formulation fromwhich any specific task can be defined. Therefore, sincesuch distinction depends on the relative number of unitsregarded as existing equipment, the three types of tasks(pure operational, retrofit and grassroots design) can beaddressed simply by pre-specifying the size (and perform-ance) for all, some or none of the units in the superstructureof configuration alternatives.

STEAM HEADER CONDITIONS

During the present work, the pressure of the headers isfixed, but the steam properties (e.g., temperatures andenthalpies) are determined before running the optimization.Note that, in contrast with previous methods, pressures andtemperatures of the steam consumed/produced by pro-cesses in each operating season might differ from theconstant conditions in the headers of the utility system.Back-pressure turbines need a certain degree of superheatto avoid wet steam in the exhaust, whereas heating/coolingsteam is normally required at near saturation conditions inorder to utilize its latent heat and exploit a higher heat transfercoefficients. After steam header pressures of the utility planthave been set, the process of establishing their appropriatetemperature levels can be divided in the following steps:

(1) Propose a temperature for the lowest-pressure header.(2) Simulate a condensing turbine of the biggest size avail-

able (i.e., the maximum limit specified for the optimiz-ation) at full-load that takes steam from this header.

(3) Check if the dryness in the exhaust of this condensingturbine is above the minimum value to avoid bladedamage (typically 90%).

(4) If not, increase the superheat degree of the LP headeruntil exhaust dryness meets the constraint.

(5) Propose a temperature for the next higher-pressureheader.

(6) Simulate a back-pressure steam turbine of the biggestsize available at full-load between the two headers.

Figure 2. Diagram of an existing utility plant showing some possible retrofit options.


1152 AGUILAR et al.

(7) Check if the exhaust of the turbine coincides with thetemperature of the lowest-pressure header

(8) Repeat steps 5–7 until convergence.(9) Apply steps 5–8 to the rest of the headers in pressure-

ascending order.(10) Once the final temperatures have been determined,

calculate header enthalpies and entropies.

By applying this iterative procedure it is possible to guaran-tee that the dryness in the discharges from condensingturbines will not go below allowed limits. Furthermore, theexhaust of back-pressure units will be at or slightly abovethe corresponding header temperature. Consequently,

letdown steam is not required to compensate for temperaturedifferences and, hence, no expansion path will be biased frominefficient operation (i.e., all paths have the same opportunityto be selected by the optimization). Thus, although steamfrom a turbine cannot be discharged at lower temperaturethan the headers, it might be rejected at higher temperaturesif the size of the unit is considerably smaller than themaximumlimit and/or if it operates at low partial loads. However, even insuch cases, it is only necessary to inject a small amount offeed water to balance the temperature of the header with neg-ligible economic impact. Note that once the final steam tur-bines have been selected (i.e., after the optimization) it isalways possible to manually tune header temperatures inorder to completely eliminate any de-superheating water injec-tion. In addition, it is interesting to note that, by following thesuggested steam temperature selection procedure, the oper-ating curve of the system on an enthalpy versus entropygraph (see Figure 3) closely resembles the ones observed inreal utility plants, which have been normally obtained throughmany years of experience.

SUPERSTRUCTURE OF CONFIGURATIONALTERNATIVES

As explained in the previous section, when solving a grass-roots or retrofit design problem the optimization proceduremust determine the units to be installed in the utilitysystem. Types and connections for new equipment representstructural decisions that, given the nature of the mathematicalsearch routine, must be taken from a finite number of optionsin a superstructure of decisions. Figure 4 illustrates an

Figure 3. Mollier chart of the expansion line for the steam headerswithin a typical utility system (very-high, high, medium, low andvacuum pressure levels).

Figure 4. Example of a superstructure of configuration alternatives from which the optimization routine can select the new equipment units to beinstalled in the utility system.



example of a superstructure in which, for instance, it is poss-ible to select up to three boilers and two GTþHRSG produ-cing steam at VHP level. Also, there are several choices forback-pressure and condensing steam turbines operating aselectric generators or direct mechanical drivers (e.g.,attached to pumps or compressors).While it is necessary to define a superstructure whenever

mathematical optimization techniques are applied to designproblems (i.e., involving configuration changes), in the presentwork there are two major distinctions with previousapproaches. First, the size of each piece of equipment rep-resented in the superstructure is an (unknown) variable to bedetermined by the optimization. In this sense, every unit rep-resents a continuous range (i.e., an infinite number) of sizeoptions within the practical limits previously established (asinput data) by users. The major advantage of this feature isthat the search space is not restricted to equipment of fixedsize and, hence, an extensive number of choices can be eval-uated without having to define many units of discrete sizescovering a certain range, which would incur in large compu-tational penalties (i.e., as more binary variables are required).Second, with the proposedmethodology, users can customizethe superstructure to represent specific plant arrangements.Hence, it is possible not only to add or remove units, butalso to fix the size of some or even all the components of thesuperstructure in order to address grassroots, retrofit and(pure) operational problems with a common optimization fra-mework. Finally, it should be noted that, given the robustnessof the linear (MILP) optimization routine, a large number ofunits might be defined in the superstructure.

OBJECTIVE FUNCTION

Mathematical programming algorithms involve the minimiz-ation (or maximization) of an objective function. The aim is tooffer a systematic way of generating optimum solutions fromwhich users can better support their decisions.Operating cost: This is the overall cost of operating the uti-

lity system through the whole time horizon being considered(e.g., $/year) and corresponds to the expense that must bemade on a continuous basis to keep the plant running. Inthe proposed study, operating costs for fuel [equation (2)],water [i.e., demineralized and cooling water, equation (3)],electricity [i.e., export or import, equation (4)], emissions[charges or credits, equation (5)] and other operatingexpenses (e.g., maintenance) are considered.

OpCst ¼ (FuelCst þ PowCst þWatCst

þ EmmCst) � Fopcst þ FixOpCst (1)

FuelCst ¼Xt

(C f1t �Mtot�f1

t � ft) � hrstot

þXt

(Cf2t �Mtot�f2

t � ft) � hrstot þ � � �

þXt

(Cfmt �Mtot�fm

t � ft) � hrstot (2)

WatCst ¼Xt

(Cdwt �Mdw


þXt

(Ccwt �Mcw


PowCst ¼Xt

(Cimpt �Weimp


�Xt

(Cexpt �Weexpt � ft) � hrs

tot

þ Cimpfix or Celec �

Xt

(Weimpt � ft)

�Xt

(Weexpt � ft)

!� hrstot þ Cimp

fix (4)

EmissCst ¼Xt

(CCO2t �Mover�CO2


þXt

(CNOxt �Mover�NOz


Capital cost:

CapCst ¼ Fcepci � Finst �Xn

PurchCstn

" #þ Capfix

(6)

PurchCstn ¼ f (Sizen) (7)

Total annualized cost: An annualization factor is employedto obtain total annualized cost from capital and operatingcosts as shown in equation (8).

TotCst ¼ OpCst þ CapCst � Fann (8)

Overall emissions: These represent the sum of CO2, NOxand SOx released to the atmosphere by the utility systemthrough the entire time horizon (e.g., ton/year). As shownin equations (10) (11), total emissions are obtained byadding the contributions from an external power source (ifelectricity is imported) and the several fuels being consumedinside combustion units within the plant in each operatingscenario.

TotEmiss ¼ hyr �Xt

(Mtot�CO2t þMtot�NOz

t ) � ft (9)

Mtot�CO2t ¼

Xn

FCO2f1 �M f1

n,t

�þ FCO2

f2 �M f2n,t

þ � � �FCO2fm �M fm

n,t

�þ FCO2

ext �Weimpt (10)

Mtot�NOxt ¼

Xn

FNOxf1,n �M f1

n,t þ FNOxf2,n �Mf2

n,t

þ � � �FNOxfm,n �M fm

n,t þ FNOxext �Weimp

t (11)

Overall electricity output: This variable corresponds to theelectricity produced in-site by the utility system (i.e., throughturbine-driven generators) for the entire time horizon. It is cal-culated by adding the weighted contributions from all gener-ating units in each time period and multiplying the sum bythe total operating hours [see equation (12)].

TotPow ¼Xn

Xt

(W elecn,t � ft) � hrs

tot (12)

MATHEMATICAL PROBLEM STATEMENT

The multi-period mixed-integer linear programming (MILP)problem for the suggested methodology can be stated as:


1154 AGUILAR et al.

Objective:

minObjFunc ¼ f (xn, yn,t, Zseln , Zop

n,t ) (13)

subject to

hq(xn, yn,t, Zseln , Zop

n,t ) ¼ 0, q ¼ 1, 2, . . . , Q

gt(xn, yn,t, Zseln , Zop

n,t ) � 0, r ¼ 1, 2, . . . , R

xn [ {xminn � xn � xmax

n } Zseln [ {1, 0}

yn,t [ {yminn;t � yn;t � ymax

n,t } Zopn,t [ {1, 0}

(14)

From equations (13) and (14), in the proposed method-ology, both maximum and actual outputs (i.e., equipmentsizes and loads) are considered (unknown) variables to beoptimized simultaneously. Therefore, in contrast with conven-tional approaches where operational and synthesis issuesare de-coupled for simplification purposes, in the presentwork the design and operation of utility systems aremerged into a single optimization task for which the solutionspace is much larger. The mathematical constraints incorpor-ated in the optimization framework developed during thesuggested approach can be classified into the following:

Electrical Balances

In order to accurately calculate the charges or credits dueto electric power entering or leaving the utility system it isnecessary to establish a balance between the electricitysources and sinks (for all operating scenarios) through aseries of equality constraints represented by equation (15).Thus, while the right side of this expression accounts forthe supply of electricity (i.e., on-site generation plus importedpower), the terms of the left correspond to the potential con-sumers, namely, process electricity demands, exported elec-tricity, electricity losses and auxiliary electric power. Asindicated by equation (16), auxiliary power comprises of theelectricity consumed by feed water and cooling systempumps, fans inside boilers and cooling towers, HRSG (inthe form of electric penalties imposed to GT generatorswith heat recovery units), and electric motors driving largerotational equipment.

Wedem þWeexp þWeaux þWeloss ¼ Wegen þWeimp

(15)

Weaux ¼ Wepmpbfw þWepmp

cw þWefancw þ

XWefan

boi

þX

WepenHR þ

XWeEMdrvs (16)

Mass Balances

Within utility systems there are many types of streamsbeing mixed in numerous connections and, in all cases, thesum of flows entering a given node must equal the totalmass leaving [equation (17)]. Equation (18) represents themass balance for the deaerator, and equation (19) corre-sponds to a generic mass balance around steam headers.Since in the current approach it is assumed that processescannot return more condensate than the total water/steamprovided by the utility plant, the makeup water must beequal [as stated in equation (20)] to the condensate lost byprocesses, together with the losses of the utility plant (i.e.,steam injected to GT, boiler and HRSG blowdowns, vents

and distribution leakages).XMin ¼

XMout (17)

Mstmdea þMret þMcond þMmkup ¼ Mbfw þMvnt

dea (18)Xk

Mboik þ

Xk

MHRk þ

Xk

Mgenk þ

Xk

MST�ink

þXk

M let�ink þMdsh

k ¼Xk

Mconsk þ

Xk

MST�outk

þXk

M let�outk þ

Xk

Mvntk (19)

Mmkup ¼ Mboibldwn þMHR

bldwn þMinjGT þ

Xk

Mvntk

þXk

(Mconsk �Mgen

k )þMret þMvntdea þMdist

loss (20)

Heat Balances

Whenever two or more stream flows at different conditions(e.g., temperature) are adiabatically mixed, a balance isneeded to ensure that the total amount of enthalpy enteringthe node equals that leaving [i.e., equation (22)]. Two majorheat balances are represented: first, in equation (23), sinceall the enthalpies depend on pre-specified data (e.g., deaera-tor pressure), the equation guarantees that enough steam isinjected into the deaerator so that the boiler feedwater leav-ing this unit is at saturated liquid conditions. Second, the bal-ance [i.e., equation (24)] ensures that the requiredtemperature inside each steam header can be maintainedby regulating the injection of letdown steam and/or de-super-heating water given that the enthalpy of steam turbineexhausts might vary for different time periods. This equationcan be simplified when some of the entering streams (e.g.,from boilers and HRSG) are already at the (pre-specified)temperature of the header [i.e., equation (25)]. Furthermore,note that (according to the modelling approach presented inthe first part of this paper) the terms representing the heatdischarged by steam turbines do not involve an explicit calcu-lation of the (specific) exhaust enthalpies (i.e., bilinear termsin the form of h exh . M exh). Instead, the net heat flows fromthe turbines are employed (i.e., Qexh in kW, which is equival-ent to the bilinear terms) and, in this way, it is possible to keepall expressions linear.X

Qin ¼X

Qout (21)X(hin �Min)¼

X(hout �Mout) (22)

hstmdea �Mstmdeaþhret �Mret þhcond �Mcond

þhmkup �Mmkup ¼ hfdea �M

bfwdeaþhgdea �M

vntdea (23)X

k

(hhdrk �Mboik )þ

Xk

(hhdrk �MbHRk )þ

Xk

(hgenk �Mgenk )

þXk

(hST�ink �MST�in

k )þXk

(hlet�ink �M let�in

k )

þXk

(hbfwk �Mdshk )þ

Xk

(hhdrk �Mvntk )

¼ hhdrk

Xk

Mboik þ

Xk

MbHRk þ

Xk

Mgenk

þXk

MST�outk þ

Xk

M let�ink þ

Xk

Mdshk þ

Xk

Mvntk

!(24)



Xk

(hgenk �Mgenk )þ

Xk

QST�ink þ

Xk

(hlet�ink �M let�in

k )

þhbfwk �Mdshk ¼ hhdrk �

Xk

Mgenk þ

Xk

MST�ink

þXk

M let�ink þMdsh

k

!(25)

Equipment Limits

A series of constraints are considered in equations (26) to(33) to reflect physical and practical limitations of the equip-ment. The first two constraints [i.e., equations (26) and(27)] limit a (continuous) size range for each potential pieceof equipment available in the configuration superstructure.Note that, equipment design outputs do not vary with time,but actual output might be different for several scenarios.Also, since it is a function of both actual input and size, if aunit is momentarily switched off, the design output should(mathematically) become 0. Consequently, the next pair ofconstraints [equations (28) and (29)] defines a ‘virtual size’that must also comply with the same maximum/minimumlimits, but which has an operational binary variable associ-ated to it. In this way, according to equations (30) and (31),any virtual size must be equal to its corresponding permanentvalue when the units are operating, whereas they becomezero if the equipment is off.Additionally, equation (32) imposes the actual output to be

lower than both, permanent and virtual sizes, ensuring thatthe equipment does not exceed its design capabilities. More-over, as indicated by equation (33), it is possible to define aminimum partial load for different types of units so that theratio of actual output to the maximum does not go below agiven specified fraction, unless the unit is switched off.

OutputDn � Limlown � Zsel

n (26)

OutputDn � Limupn � Zsel

n (27)

OutputVDn,t � Limlown � Zop

n,t (28)

OutputVDn,t � Limupn � Zop

n,t (29)

OutputDn � OutputVDn,t (30)

OutputDn � OutputVDn,t þ Limupn � (1� Zop

n,t ) (31)

OutputVDn,t � Outputn,t (32)

Outputn,t � Loadminn �OutputVDn,t (33)

Equipment Performance Equations

Each item within the utility plant must also meet certainperformance equations. Such mathematical expressionshave been derived for different types of equipment (i.e.,multi-fuel boilers, steam turbines, gas turbines, HRSG) andwere presented in the first part of this paper (Aguilar et al.,2007).

MECHANICAL DRIVER SELECTION

In the context of industrial energy systems, mechanical dri-vers, or prime movers, are items of equipment directly satis-fying shaft power requirements for a processing plant. Suchunits convert other types of energy into mechanical powerand are attached to rotational devices such as pumps,

compressors, fans, conveyors and mixers. Note that unitsmoving electric generators are normally considered electricityproducers rather than mechanical drivers because they arenot directly providing shaft power to any processing equip-ment. While in many cases the specifications of the demandsimpose a particular type of driving equipment to be installed,whenever there are degrees of freedom in choosing the dri-vers, such alternatives can be exploited to improve the over-all economics of the utility plant. Furthermore, as indicated byTable 1, although most of the shaft demands in a typical siteare too small for being included in the driver selection (i.e.,only electric motors are economically feasible), units above250 kW represent the largest fraction of the energy consump-tion and cost requirements. Consequently, it is possible toachieve significant economic savings if mechanical driversare properly selected.The first question that must be answered during the driver

selection procedure is what types of units should beinstalled. Electric motors, steam turbines, gas turbines andinternal combustion engine are four main groups of primemovers commonly found in industrial applications, eachoffering different comparative advantages and drawbacks.As shown in Figure 5, each of these major types of drivingequipment can be incorporated into the utility system inseveral ways. For instance, gas turbines with HRSG mightdeliver steam at different levels, whilst steam turbines(back-pressure and condensing) can be installed betweenmany expansion paths, and motors might consume electri-city generated on-site or imported from external suppliers.Clearly, driver selection can have a considerable impacton the design and operation of the utility plant. It can there-fore be exploited to reduce capital and/or operating costs.However, the number of alternatives grows exponentiallyas more variables are taken into account. Thus, apartfrom types and placement (i.e., connections within the utilitysystem) of the driving equipment, it is also required todecide the number and sizes of the drivers attached toeach shaft demand, and to define their loads for each oper-ating scenario.Since the design and operational degrees of freedom to

select a driver are strongly interrelated with the other vari-ables for the utility system, it is necessary to simulate theentire plant. Although this approach involves complex calcu-lations, only in this way it is possible to assess the overallconsequences of any proposed decision and determinewhether a given alternative improves the economic perform-ance of the system. Moreover, the task is also highly combi-natorial due to the multitude of (design and operational)choices to be evaluated and compared.

Table 1. Distribution of driver population and energy consumption bysize in the chemical industry (OIT DOE, 1998).

Driver size HP Driver size KW Population (%) Energy use (%)

,5 ,3.7 42.4 1.66–20 4.5–15.0 30.0 6.421–50 15.7–37.3 14.5 9.151–100 38.0–74.6 5.9 9.3101–200 75.3–149.1 4.1 14.3201–500 149.9–372.9 2.2 18.1.500 .372.9 0.9 41.2

100 100


1156 AGUILAR et al.

Despite its potential for large economic savings, driverselection is not fully exploited by many methods addressingthe design and operation of utility systems, in which all primemovers are simply pre-specified as components of externalprocesses. Thus, while these issues should be considered ina plant-wide context, they are often de-coupled from the restof the problem in order to simplify the procedure. Nevertheless,a few studies have been able to deal with the combinatorialnature of the task and its complex calculations by employingmathematical programming techniques. For instance, Papou-lias and Grossmann (1983) considered the possibility of satis-fying each shaft demand with either a single steam turbine(selected from several expansion paths within the utilityplant) or an electric motor, both of pre-specified sizes. Later,Maia and Qassim (1997) extended this concept to multi-period problems in which the steam turbine or electric motor(of pre-specified size) to be attached to a fluctuating rotationaldemand must operate at part-load during some scenarios.More recently, Del Nogal et al. (2003) increased the numberof driving equipment options (e.g., gas turbines) and includedthe possibility of several drivers moving a common shaft tomeet the base-load requirements of power-dominated pro-cesses.1 Conversely, the present work considers an extensivenumber of driver alternatives from which the driving units

attached to each shaft demand along with their sizes andloads (in each operating scenario) are variables within the con-text of an integrated methodology to optimise the design andoperation of flexible utility systems.The first step to address driver selection within the present

methodology is to incorporate an extensive number of drivingequipment options to satisfy a given set of shaft demandsthat might fluctuate in different time periods. As shown inFigure 6, each shaft can be driven by several electricmotors, gas turbines with or without HRSG, back-pressureor condensing steam turbines, or by a combination of them.Note that the number of drivers attached to a single shaftand their sizes are (unknown) variables whose value is deter-mined by the optimization routine. In this way, the types andnumber of drivers attached to each shaft demand, their sizesand loads, in each operation scenario are optimised, togetherwith the entire utility system.Additional constraints, from equations (34) to (41), which

reflect the physical and practical limitations of the driving

Figure 5. Several types of mechanical drivers can be placed within the energy system, consuming and/or delivering utilities.

1Since in such processes there are usually many (compatible) com-pression units running at the same time, in this approach it is alsopossible to merge/split some of the shaft demands to distributethem among less/more driving units.



units are required in the optimization framework. In this casethe variables must be identified not only by type of equipment(index j), but also by the shaft to which the driver is attached(index i). Additionally, equation (42) guarantees that theactual power delivered by all the drivers attached to acommon shaft meets the corresponding demands in eachoperating scenario.

WdrvDi,j � Limlowj � Zsel

i,j (34)

WdrvDi,j � Limupj � Zsel

i,j (35)

WdrvVDi,j,t � Limlowj � Zop

i,j,t (36)

WdrvVDi,j,t � Limlowj � Zop

i,j,t (37)

WdrvDi,j � WdrvVDi,j,t (38)

WdrvDi,j � WdrvVDi,j,t þ Limupj � (1� Zop

i,j,t) (39)

WdrvVDi,j,t � Wdrvi,j,t (40)

Wdrvi,j,t � Loadminj �WdrvVDi,j (41)

Wshafti,t þ ai � Vart ¼Xj

Wdrvi,j,t (42)

CASE STUDIES

In order to demonstrate the applicability of the suggestedmethod, this section describes several analyses carried outfor an existing central utility system that is facing a major ret-rofit, given that a large process will be added to the site andthe energy requirements are expected to increase consider-ably. The objective of the problem is to determine not onlythe new equipment (types, number of units, sizes and con-nections) to be installed, but also the operating conditionsof the whole utility system that would represent the most cost-effective way of meeting the expected demands. The mainsite conditions for this case are presented on Table 2,where it can be observed that there are two seasons

corresponding to four summer months (33%) and eight ofbase operation (67%). During summer the variation betweenpeak and off-peak electricity tariffs is more drastic and thenumber of peak hours per day is also larger in comparisonto the base season. In addition, Table 3 shows that existingsteam and mechanical power needs are higher for the basescenario, whereas the required electricity is lower than insummer. Moreover, although future demands will follow simi-lar trends, the utility system should produce around 30%more electricity and 40% of additional mechanical powerand steam.On the other hand, from Figure 7, it is possible to identify

four boilers of 40 kg s21, three of which discharge steam atVHP level and the other one to the HP header. Also, fiveback-pressure turbines (four between VHP-HP, plus onebetween HP-LP) of different sizes expand most of thesteam to produce electricity before satisfying the heating

Figure 6. An exhaustive number of driver choices (of unknown size) are included in the superstructure of configuration alternatives.

Table 2. Site data for the case studies.

Global site conditions

Total working hours hrs/year 8600Altitude m 0Fuel oil #2 LHV kJ/kg 45000Natural gas LHV kJ/kg 50244

Site conditions per seasonSeason Base SummerFraction of the year % 67 33Ambient temperature 8C 10 25Relative humidity % 60 60Electric prices: Peak $/kWh 0.07 0.08Off-peak $/kWh 0.05 0.05Peak hours per day hrs 7 12Fuel oil #2 prices $/kg 0.19 0.19Natural gas prices $/kg 0.22 0.22Raw water prices $/ton 0.05 0.05


1158 AGUILAR et al.

needs of the users. An additional condensing steam turbo-generator is also installed between MP and the vacuum con-denser to extract power from any extra steam not consumedby the processes. The corresponding water/steam press-ures and temperatures are summarized in Table 4. Regard-ing the mechanical demands, there are four large pumps,each of which has a steam turbo-driver and an electricmotor (both of the same size) attached to its shaft (thefourth one is actually representing two units lumpedtogether). Even though the utility plant has some redundantcapacity with several units installed in parallel (i.e., perform-ing the same function), they will not be enough to satisfy theadditional requirements and, hence, new equipment must beprovided.

Operational Optimization of the Existing Plant

As an initial task, the proposed approach can be employedto check if there is scope of achieving some savings byimproving the operation of the existing system. Therefore, itis first necessary to simulate the utility plant under its currentoperational policy, and then to optimize it considering overallcost as the objective to be minimized. Note that at this pointno design modifications are considered since there is noinvestment available. Also, for comparative purposes, vari-able conditions and demands will be handled by partitioningthe time horizon in two seasonal intervals (base andsummer) and further sub-dividing each of them in two inter-seasonal periods corresponding to peak and off-peak electri-city prices (i.e., four scenarios in total).Figure 7 shows the operational data of the plant during

the base season (summer operation is similar and hasbeen omitted for simplification purposes). Under the currentpolicy there are no operational differences between peak

Table 4. Steam conditions for the case studies.

Steam header conditions

VHP pressure bara 101HP pressure bara 20.6MP pressure bara 4.1LP pressure bara 2.7Deaerator pressure bara 1.1Condenser pressure bara 0.98VHP temperature 8C 539HP temperature 8C 333MP temperature 8C 186LP temperature 8C 150Condensate temperature 8C 90

Figure 7. Flowsheet for the existing plant (base season).

Table 3. Demands data for the case studies.

Requirements for the utility system

Existing

FutureBase Summer Base Summer

Electricity demands MW 62 68 81 89VHP steam demands MW 112 101 157 141HP steam demands MW 200 180 280 252MP steam demands MW 42 38 58 53LP steam demands MW 70 63 98 88Total steam demands MW 424 382 593 534Condensate return % 80 80 80 80Power pump 1 MW 5.2 5.0 7.2 7.0Power pump 2 MW 1.3 1.1 1.8 1.5Power pump 3 MW 2.2 2.0 2.8 3.0Power pump 4 MW 0.6 0.6 0.8 0.8Process CW demands MW 200 300 280 420



and off-peak periods and VHP boilers, together with thecondensing turbine tend to be operated almost at its maxi-mum load. Also, the letdown through the first valve isalways zero and the steam flow between VHP and HPheaders is evenly distributed among the corresponding(four) turbo-generators. However, although the turbodriversare running at their maximum capacities (i.e., electricmotors kept as passive back-up units), there is still a con-siderable amount of steam being expanded through thesecond and third letdown valves in order to meet thedemands at MP and LP levels. Additionally, despite thespare generation capacity, it is not possible to producemore electricity given that steam venting is not allowedand that the condensing turbine is already at full load. Inthis way, as presented in Table 5, fuel consumption andelectricity purchases represent the two major expensessince the utility system can only generate around 45% ofthe required electric power.Figure 8 presents the optimized flowsheet for the base

season. The mathematical model consisted of 3152equations, 1892 variables, of which 218 were binary, and ittook 136 major iterations to arrive at a solution with 0.005%of relative gap (GAMSw 2.0, Cplex solver; Brooke et al.,1998) in 0.151 s of CPU time2 (Pentium IV processor at3.0 GHz). In this particular case, no operating differencesbetween peak and off-peak periods were obtained, sincemajor units cannot be started up quickly enough for thepeak-periods and because of the strong drive to reduce elec-tricity imports (i.e., it is not possible to export power and it isnot cost-effectively to increase import). Also, similar to thenon-optimized situation, VHP boilers and the turbo-driverstend to be operated at full capacity. However, in contrastwith the previous policy, one of the VHP-HP steam turbinesis shut down so that the other three run almost at their maxi-mum capacities. In this way, not only the first letdown is stillzero, but also the flow through the second valve is signifi-cantly lower and, hence, the condensing turbine can beswitched off. The turbo-drivers are also operated at theirmaximum capacity since there are no other alternatives tofurther exploit the cogeneration potential (i.e., to cut letdownflows even more).As a result of such measures (see Table 5), the overall

operating cost of the utility system has been reduced by$3.73 millions year21 (4.3%) mainly due to savings in fuelconsumption. Consequently, even though operating the con-densing turbine at full load was supposed to be de-bottle-necking electricity production from back-pressure units, theinstalled power-generation capacity is not enough to fullyexploit the cogeneration potential for the heating needs asthe letdown flows cannot be eliminated. Moreover, by shut-ting down two of the turbines and running the rest (including

the drivers) at their full capacities (i.e., at higher loads), notonly has it been possible to extract practically the sameamount of power, but also to save some steam (and fuel)from the HP boiler. Furthermore, the task becomes evenmore challenging when design options are incorporatedand, as illustrated in the following examples, it is also necess-ary to employ an integrated approach to deal with suchissues.

Conventional Retrofit by Inspection (Base Case)

Any retrofit proposal for a utility plant should take intoaccount different operating scenarios to ensure that thefinal configuration is capable of meeting the expecteddemands under all conditions. With the use of conventionaltechnologies it is necessary to screen out any design optionsby inspection or heuristic rules. In such cases the only possi-bility is to evaluate the economic performance of severalpromising alternatives with an operational optimizer andthen to adjust some equipment sizes by trial and error.Clearly, in this type of approach, not only are design andoperating issues not considered at the same time, but alsothe configuration options are limited to those initially selectedby users.In order to illustrate the advantages of applying the meth-

odology developed during the present work, a retrofit optionobtained with conventional procedures (i.e., proposing thedesign alternatives by inspection) will be used as a referencecase. This alternative is shown in Figure 9, where it is poss-ible to observe three new VHP boilers of the same size as theexisting ones (40 kg s21). Also, two additional steam turbinesgenerating electricity and four new turbo-drivers were pro-posed to exploit the cogeneration potential of the extrasteam required at lower levels. Sizes and expansion pathsfor the generators were selected so that their rated massflows could supply the increased heating demands withoutthe need of any steam passing through letdown valves. Simi-larly, the new drivers were sized to meet the minimumexpected shaft demands on their own (i.e., during summer)and they were placed between VHP and HP headers sothat they can also employ the steam discharged by upstreamturbines.This system was optimised operationally with the proposed

framework by considering 12 seasons (i.e., months) and twointer-seasonal intervals for each of them (i.e., peak and off-peak) which follow the same relative time distribution indi-cated originally (i.e., four summer months and the rest,67%, base demands). The mathematical model consistedof 21 550 equations, 10 835 variables, of which 506 werebinary, and it took 440 720 major iterations to arrive at a sol-ution with 0.27% of relative gap (GAMSw 2.0, Cplex solver,Brooke et al., 1998) in 235 s of CPU time (Pentium IV pro-cessor at 3.0 GHz).Figure 9 presents the optimum flowsheet for a particular

season, which can be used to illustrate the general trendsobserved in other periods. Again, just small operating differ-ences between peak and off-peak intervals were obtaineddue to the strong drive to reduce electricity import and therestriction of starting major units just for the peak hours. In

Table 5. Cost comparison between the current and optimizedoperation of the existing system.

Cost comparison

Units Current Optimal Difference

Overall operating cost $106/year 85.94 82.21 23.73Overall fuel cost $106/year 65.76 62.23 23.53Overall water cost $106/year 0.37 0.32 20.05Overall electricity cost $106/year 19.81 19.66 20.15

2Since this is only an operational optimization task, it has beenpossible to achieve a relatively low computational time.


1160 AGUILAR et al.

this case, boilers and generators also tend to be operated attheir maximum capacity so that letdown steam can be almosteliminated and the condensing turbine turned down. Also,since there are other alternatives to exploit cogenerationpotential (i.e., additional back-pressure turbines), turbo-dri-vers are not always running at full-load and electric motors

are sharing some of the load for the pumps during certainperiods. The site is producing around 60% of the requiredelectricity and so, as can be appreciated in Table 6, the rela-tive cost of electricity is lower compared to the originalsystem. On the other hand, the required investment for thisretrofit option is $62.0 million corresponding to an overall

Figure 9. Flowsheet for the retrofit option obtained by inspection (January off-peak period).

Figure 8. Operational flowsheet for the (optimized) existing plant (base season).



operating cost of $111.0 millions year21. As explainedbefore, this design alternative obtained by inspection (butoptimized operationally with the suggested framework) willbe employed as reference case to demonstrate the capabili-ties of the proposed methodology.

Minimum-Investment Retrofit Option

The proposed approach has been employed to first deter-mine the minimum-investment option, by selecting overalloperating cost as objective function and reducing the avail-able budget until no feasible solution was found. Again, inthis optimization the time horizon was partitioned into 12 sea-sonal periods further sub-divided in two inter-seasonal ones(i.e., 24 scenarios). The optimization model consisted of19 676 equations, 9929 variables, of which 552 werebinary, and it took 4 212 056 major iterations to arrive at a sol-ution with 0.36% of relative gap (GAMSw 2.0, Cplex solver;Brooke et al., 1998) in 1542 s of CPU time (Pentium IVprocessor at 3.0 GHz).

From Figure 10, it can be observed that, similar to the ret-rofit by inspection, there are three new VHP boilers (two of40 and one of 36.4 kg s21). Also, additional back-pressureturbo-drivers were supplied to the pumps, but their sizesare smaller and their expansion paths are between VHPand LP (rather than employing HP steam as in the previousoption). Note that, while boilers and electric generators tendto be operated at their maximum capacities, the plant is notcondensing any steam. Also, even though shaft demandsare being satisfied only by the turbo-generators (new andexisting sharing the load, leaving electric motors as passiveback-up), letdown flows are much higher than in previouscases since no additional turbines can exploit the cogenera-tion potential for the increased demands. The requiredinvestment for this option, as shown in Table 7, is only$29.0 million, representing savings for $33.0 million (46%reduction) compared to the retrofit by inspection. Neverthe-less, the overall operating cost is slightly higher (115.3against $111.0 millions year21) due to the additional electri-city being purchased which represents around 57% of thetotal requirements.

Figure 10. Flowsheet for the minimum-investment retrofit option obtained with the proposed approach (January off-peak period).

Table 7. Major economic parameters for the minimum-investment retrofit option obtained with the proposedapproach.

Economic parameters

Units Current

Required investment $106 29.0Overall operating cost $106/year 115.1Overall fuel cost $106/year 85.1Overall electricity cost $106/year 29.3Overall water cost $106/year 0.68

Table 6. Major economic parameters for the retrofit byinspection.

Economic parameters

Units Current

Required Investment $106 62.0Overall operating cost $106/year 111.0Overall fuel cost $106/year 87.5Overall electricity cost $106/year 22.8Overall water cost $106/year 0.68


1162 AGUILAR et al.

Other Retrofit Options with AdditionalInvestment

Once the minimum-investment retrofit has been deter-mined, the suggested approach can be employed to investi-gate other options with higher investment. Thereafter, it willbe first considered that it is possible to invest an extra $10million (i.e., a total of $39 million), assuming constraints forthe same number of periods than before. The correspondingoptimization task consisted of 19 676 equations, 9977 vari-ables, of which 646 were binary, and it took 11 626 593major iterations to arrive at a solution with 1.6% of relativegap (GAMSw 2.0, Cplex solver; Brooke et al., 1998) in9245 s of CPU time (Pentium IV processor at 3.0 GHz).From Figure 11 it is possible to observe that there are three

new VHP boilers almost of the same size as in the minimum-investment case (two of 40 and one of 38.5 kg s21) given thatthe main difference between both solutions (for this particularsituation) does not rely on the steam supplied (amount andlevel), but on how it is expanded before being delivered tousers. Thereafter, two additional turbo-generators of11.1 MW and 15.2 MW have been installed (between theVHP-LP and VHP-MP headers) and four new turbo-driversbetween VHP-LP levels have been provided for the pumpsin order to exploit better the cogeneration potential. Notethat, although the sizes of the new drivers are slightlybigger compared to the previous retrofit option, their expan-sion path is the same as before since it corresponds to thelowest capital per kW.Given that boilers and steam generators tend to be oper-

ated at their full capacities (i.e., at their highest efficiencies),flows through the first two letdown valves have been elimi-nated and the third one drastically reduced so that morepower can be extracted from the steam before meeting the

heating demands at lower levels (i.e., producing around70% of the required electricity). Therefore, since there aremore possibilities to efficiently expand the steam (includingdrivers of bigger sizes), electric motors can be loaded formore periods without incurring high cost penalties. In thisway, as shown on Table 8, the overall operating cost of theplant has been reduced to $108.1 millions year21, represent-ing savings of $7.0 millions year21 (6.1%) and a paybacktime of 1.43 years, taking as reference the minimum-invest-ment option. Moreover, compared to the retrofit by inspection,this solution also features investment savings for $23 million(37.1%) and a reduction of $2.9 millions year21 (2.6%) inoperating costs.Even though this retrofit solution is clearly more cost-effec-

tive than the one obtained by inspection, it might still be poss-ible to improve these results (e.g., to reduce payback timefrom the minimum-investment option) by considering ahigher investment. Consequently, for the next example an

Figure 11. Flowsheet for the retrofit option obtained with the proposed approach assuming $10.0 million of additional investment (Januaryoff-peak period).

Table 8. Major economic parameters for the retrofit option obtainedwith the proposed approach, assuming $10.0 million of additionalinvestment.

Economic parameters

Units Current

Required investment $106 39.0Overall operating cost $106/year 108.1Overall fuel cost $106/year 88.7Overall electricity cost $106/year 18.7Overall water cost $106/year 0.68Savings from minimum investment option $106/year 7.0Payback time year 1.43



additional investment of $20.0 million (i.e., a total of $49.0million) has been allowed. The optimization model for thiscase consisted of 19 676 equations, 9974 variables, ofwhich 646 were binary, and it took 16 421 014 major iter-ations to arrive at a solution with 1.84% of relative gap(GAMSw 2.0, Cplex solver; Brooke et al., 1998) in 13 845 s

of CPU time (Pentium IV processor at 3.0 GHz). In thiscase the optimal operational flowsheet for one representativescenario is presented in Figure 12 and, as can be observed,two boilers of 40.0 kg s21 and a 25.6 MW gas turbine witha supplementary-fired HRSG of 23.3 kg s21 have beeninstalled to supply the required steam. Note that, althoughthis additional steam production capacity is lower comparedto the previous example, now the utilisation factor is largeras the HP boiler is used for more periods. Thereafter, sincethere is less steam available in the VHP header, only one13.6 MW back-pressure turbine has been placed betweenVHP-LP levels. Therefore, the flows through the letdownvalves are higher than in the last case. Nevertheless, morepower can be generated as a whole with the new GT, andnow around 90% of the electricity requirements are satisfiedby the utility system. Also, given that more capital has beenassigned to the electric generators, sizes for the new turbo-drivers are slightly smaller than in the $10 million case.Due to the large quantities of steam to be expanded, thenew and old turbo-drivers are meeting shaft requirementsfor most of the time (leaving the electric motors as back-upunits). In this way (see Table 9) the overall operating cost

Figure 12. Flowsheet for the retrofit option obtained with the proposed approach assuming $20.0 million of additional investment (Januaryoff-peak period).

Table 9. Major economic parameters for the retrofit option obtainedwith the proposed approach, assuming $20.0 million of additionalinvestment.

Economic parameters

Units Current

Required investment $106 49.0Overall operating cost $106/year 103.6Overall fuel cost $106/year 90.7Overall electricity cost $106/year 12.2Overall water cost $106/year 0.68Savings from minimum investment option $106/year 11.5Payback time year 1.74

Table 10. Steam production from boilers for the additional $10.0 million retrofit option (off-peak scenarios).

kg s21 Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

VHP BO 1 32.8 0 40 40 40 40 40 40 40 40 32.8 40VHP BO 2 40 0 40 32.6 40 22.4 40 22.4 32.6 32.6 40 40VHP BO 3 12.0 40 32.8 12.0 21.3 0 23.4 40 40 40 40 40VHP BO 4 40 40 12.0 40 40 40 0 40 40 40 40 32.5VHP BO 5 40 40 40 40 40 40 40 0 12.0 12.0 12.0 12.0VHP BO 6 38.5 38.5 38.5 38.5 0 38.5 38.5 38.5 38.5 38.5 38.5 38.5HP BO 1 0 40 0 0 0 0 0 0 0 0 0 0


1164 AGUILAR et al.

Table 11. Power generation from steam turbines for the additional $10.0 million retrofit option (off-peak scenarios).

MWe Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

VHP-HP ST 1 7.0 7.0 7.0 7.0 7.0 7.0 0.0 7.0 7.0 7.0 7.0 7.0VHP-HP ST 2 10.0 10.0 10.0 10.0 10.0 0.0 10.0 10.0 10.0 10.0 10.0 8.7VHP-HP ST 3 9.7 0.0 9.7 10.0 10.0 10.0 10.0 10.0 9.6 9.6 9.7 10.0VHP-HP ST 4 10.0 10.0 10.0 9.6 10.0 10.0 10.0 0.0 10.0 10.0 10.0 10.0VHP-MP ST 1 11.1 9.9 11.1 11.1 0.0 11.1 11.1 11.1 11.1 11.1 11.1 11.1VHP-LP ST 1 15.2 0.0 15.2 15.2 15.2 15.2 15.2 15.2 15.2 15.2 15.2 15.2HP-LP ST 1 1.0 0.0 1.0 0.0 1.0 0.0 0.0 0.0 0.0 0.0 1.0 0.0MP-VP ST 1 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

Figure 13. Power from mechanical drivers through the year for the additional $10.0 million retrofit option.



of the plant has been reduced to $103.6 millions year21

representing savings of $11.5 millions year21 (10.0%) anda payback time of 1.74 years, taking as reference the mini-mum-investment option. Moreover, compared to the retrofitby inspection, this solution also features investment savingsfor $13 million (21.0%) and a reduction of $7.4 millionsyear21 (6.7%) in operating costs.To demonstrate how the operation of the utility system

varies throughout the year, Table 10 presents the steam pro-duction of the boilers in each month (off-peak period) for theextra-$10-million retrofit solution. As can be observed, all theunits are switched-off at least once per year and the HP boileris basically used as a passive back-up. Additionally, the elec-tric outputs from the steam turbo-generators are shown inTable 11, where it can be observed that they tend to be runat their maximum capacities (i.e., at their highest efficiencies).Moreover, since the system is operating in a cogenerationmode (i.e., extracting as much power from the steambefore delivering it to users) the condensing turbine (betweenMP and VP headers) remains switched-off throughout theyear. On the other hand, Figure 13 illustrates graphicallythe load distribution for the mechanical drivers through theyear. For instance, while for the first pump the new turbo-driver fully satisfies its demands for most of the time, in thelast pump the new turbine only operates during a fewmonths. Hence, not only does the resulting operation policyfor the drivers exhibit complex variations, but also it is differ-ent for each shaft demand.The main results from the four retrofit solutions in this case

study are displayed in Table 12. Clearly, despite the relativelow operating cost for the option obtained by inspection, theamount of capital required is very high. Thereafter, the mini-mum-investment retrofit determined with the suggestedapproach achieved very large capital savings (46.0%), butresulted in a slightly higher operational cost. However, if anextra $10.0 million are allowed to be invested, not only thecapital savings are still large (37.1%), but also there is nowa reduction of $2.9 millions year21 (2.6%) in the operationalcost (compared to the base case). Furthermore, if the avail-able investment is increased by another $10.0 millions (i.e.,an additional $20.0 million), the corresponding savings incapital and operating costs are $13.0 million (21.0%) and$7.4 millions year21 (6.7%). In this way, not only have theresults from the base case been improved, but also severalalternative solutions were provided, giving the opportunityto users of comparing the advantages of different economicrestrictions and/or policies.

CONCLUSIONS

The second part of this paper has explored how the numer-ous degrees of freedom involved in the design and operationof utility plants can be exploited to improve the overall

economics of these systems. However, this is a highly com-binatorial task. In the proposed methodology, mathematicalprogramming techniques are employed to effectively com-pare all the alternatives within the solution space of theseproblems. Since both design and operational parametersare optimized simultaneously, it is possible to tackle notonly grassroots design cases, but also retrofit, and pure oper-ational ones with a common framework simply by fixing someor all the elements of the superstructure of configurationchoices. Additionally, different functions can be selected asthe objective to be minimized or maximized during the sol-ution procedure of these tasks, depending on the purposeof each case. The major mathematical constraints reflectingphysical and practical restrictions of a particular problemhave been also described. Also, the concept and majorissues related to driver selection were explained, togetherwith the proposed approach to incorporate this task withinan integrated methodology to address the design and oper-ation of flexible utility systems. Finally, examples have beenpresented to demonstrate the applicability of the presentwork and its potential of achieving significant economicbenefits by improving the solutions obtained through conven-tional methods.

NOMENCLATUREai conversion factor to calculate shaft power from a

given optimization variableCtcw specific cost of cooling water makeup, $ kg21

Ctdw specific cost of demineralized water makeup, $ kg21

Ctfm cost per unit mass of the fuels fm available within the

plant in time period t, $ kg21

Ctimp specific costs of imported electric power in time

period t, $ kWh21

Cfiximp fixed connection cost for the electricity grid, $ year21

C elec specific costs of imported/exported electricityenergy at the end of the time horizon, $ kWh21

Ctexp specific costs of exported electric power in time

period t, $ kWh21

CtCO2 specific charges/credits of CO2 emissions, $ kg21

CtNOx specific charges/credits for NOx emissions, $ kg21

CapCst overall capital cost of the utility system, $Capfix fixed capital charge for the whole plant, $EmissCst overall emissions plant cost, $ year21

f1, f2. . .fm index for the several fuels consumed in the utilityplant

ft fraction of time horizon corresponding to eachperiod

Fann annualization factor to convert capital into operatingcosts

Fcepci chemical engineering plant cost index factorFextCO2 CO2 produced by external supplier per unit of

imported eletricity, kg-CO2 kWh21

FfmCO2 CO2 produced per mass unit of fuel fm in time period

t, kg-CO2 kg21

Finst installation factor to adjust equipment purchasecosts

FextNOx NOx produced by external supplier per unit of

imported electricity, kg-NOx kWh21

Table 12. Comparative table for the major results from the retrofit options of case study 1.

Inspection case Minimum investment Extra 10.0 $106 Extra 20.0 $106

Required investment $106 62.0 29.0 39.0 49.0Overall operating cost $106/year 111.0 115.1 108.1 103.6Capital cost savings $106 NA 33.0 23.0 13.0Operating cost savings $106/year NA 24.1 2.9 7.4Average power import % 40.0 43.0 30.0 10.0


1166 AGUILAR et al.

Ffm,nNOx NOx per mass unit of fuel fm burnt in unit n in time

period t, kg-NOx kg21

Fcstop factor to increase operational costs by a fixed

percentageFixOpCst fixed charge to increase overall plant operation cost,

$ year21

FuelCst overall fuel plant cost, $ year21

hin specific enthalpy of the streams entering a mixingnode, kJ kg21

hout specific enthalpy of the streams leaving a mixingnode, kJ kg21

hkbfw enthalpy of the feed water used to de-superheat

steam in header k, kJ kg21

h cond enthalpy of condensate water entering thedeaerator, kJ kg21

hdeaf enthalpy of saturated liquid at deaerator pressure,

kJ kg21

hdeag enthalpy of saturated vapour at deaerator pressure,

kJ kg21

hkgen enthalpy of the steam generated by processes and

delivered at header k, kJ kg21

hkhdr enthalpy inside each steam header, kJ kg21

hklet-in enthalpy of the letdown steam entering header k,

kg s21

h mkup enthalpy of demineralized water makeup for theutility system, kg s21

h ret enthalpy of returning condensate from processes,kJ kg21

hkST-in enthalpy of the discharges from steam turbines

entering header k, kJ kg21

hdeastm enthalpy of stripping steam entering the deaerator,

kJ kg21

hrs tot total number of hours in the time horizon (i.e.,nominal year), h year21

i index corresponding to the number of shaftdemands

Inputn,t actual input from unit n of the plant in time period tj index corresponding to types of drivers available to

satisfy power shaft demandsk index corresponding to steam headers in the utility

plantLimj

low lower limit for the design output (i.e., size) of each jtype of driver

Limjup upper limit for the design output (i.e., size) of each j

type of driverLimn

low lower limit for the design or maximum output (i.e.,size) of unit n in the plant

Limnup upper limit for the design or maximum output (i.e.,

size) of unit n in the plantLoadj

min minimum partial load for each j type of driverLoadn

min minimum partial load for unit n of the plantMin mass flow entering a mixing node, kg s21

Mout mass flow leaving a mixing node, kg s21

Mbldwnboi blowdown from all boilers within the utility system,

kg s21

Mkboi steam produced by boilers and delivered at each k

header, kg s21

M bfw boiler feed water leaving the deaerator, kg s21

M cond condensate water entering the deaerator, kg s21

Mkcons steam consumed by processes at header k, kg s21

Mtcw cooling water makeup in time period t, kg s21

Mlossdist distribution losses in the utility system, kg s21

Mkdsh de-superheating boiler feed water injected into

header k, kg s21

Mtdw demineralized water makeup in time period t, kg s21

Mn,tfn consumption of fuel fm inside unit n in time period t,

kg s21

Mkgen steam generated by processes and delivered at

header k, kg s21

MbldwnHR blowdown from all HRSG within the utility system,

kg s21

MkHR steam produced by HRSG and delivered at header

k, kg s21

MGTinj steam injected to all gas turbines in the utility

system, kg s21

Mklet-in letdown steam entering header k, kg s21

Mklet-out letdown steam leaving header k, kg s21

Mmkup demineralized water makeup for the utility system,kg s21

Mtover-CO2 overall CO2 emissions to be charged/credited in

time period t, kg s21

Mtover-NOx overall NOx emissions to be charged/credited in

time period t, kg s21

M ret returning condensate from processes, kg s21

Mdeastm stripping steam entering the deaerator, kg s21

Mttot-fn total mass consumption of fuel fm in the utility plant

in time period t, kg s21

Mttot-CO2 total CO2 emissions from the plant in time period t,

kg s21

Mttot-NOx total NOx emissions from the plant in time period t,

kg s21

MkST-in discharges from steam turbines entering header k,

kg s21

MkST-out steam to be expanded inside turbines taken from

header k, kg s21

Mdeavnt vented steam from the deaerator, kg s21

Mkvnt vented steam from header k, kg s21

n index corresponding to the units of the plantObjFunc objective function to be minimized while solving the

optimization taskOpCst overall operating plant cost $ year21

OutputnD design or maximum output (i.e., size) from unit n of

the plantOutputn,t

VD virtual design output (e.g., virtual size) from unit n ofthe plant in time period t

Outputn,t actual output from unit n of the plant in time period tPowCst overall electricity plant cost, $ year21

PurchCstn purchase cost for the n equipment unit, $q index corresponding to the equality constraints of

the problemQin heat flow entering a mixing node, kWQout heat flow leaving a mixing node, kWQk

ST-in heat flow of the discharges from steam turbinesentering header k, kW

r index corresponding to the inequality constraints ofthe problem

Sizen size of the n selected equipment unit (several units)t index corresponding to the number operating

periods of the optimization problemTotCst total annualised cost of the utility plant, $ year21

TotEmiss overall atmospheric emissions from the plant, tonyear21

TotPow overall electric energy produced by the utility system,kWh year21

Vart any optimization variable fluctuating with time period tWn,t

elec Electric power output from individual generators intime period t, kW

WatCst overall water plant cost, $ year21

Wdrvi,j,t actual output of every j driver attached to an i shaftdemand in time period t

Wdrvi,jD design or maximum output (i.e., size) for every j

driver attached to an i shaft demandWdrvi,j,t

VD virtual design output for every j driver attached to ani shaft in time period t

We aux electricity consumed by auxiliary units of the utilitysystem in each period, kWe

We dem total electricity demanded by processes in eachperiod, kWe

WedrvsEM electricity consumed by motor drivers of the utility

system in each period, kWeWe exp electricity exported by the utility system in each

period, kWeWeboi

fan electricity consumed by boiler fans in time period t,kWe

Wecwfan electricity consumed by cooling fans in time period t,

kWeWe imp electricity imported by the utility system in time

period t, kWeWe loss distribution and control electric losses of the utility

system in each period, kWeWe gen total electricity generated by the utility system in time

period t, kWeWeHR

pen electricity penalties from attaching HRSG to GTgensets in each period, kWe



Webfwpmp electricity consumed by boiler feed water pumps in

time period t, kWeWecw

pmp electricity consumed by cooling tower pumps in timeperiod t, kWe

Wshafti,t mechanical power demands for each i shaft in everytime period

xn maximum output (size) for each n unit in the plantxnmax upper limit for maximum output (size) of each n unit

in the plantxnmin lower limit for maximum output (size) of each n unit

in the plantyn,t actual output (load) of each n unit in every t

operating scenarioynmax lower and upper limits for actual output (load) of

each n unitynmin lower and upper limits for actual output (load) of

each n unitzn,top binary variables for the on/off status of each n unit in

operating scenario tznsel binary variables to select equipment unitszi,jsel binary variables to attach (select) a j type of driver to

an i shaft demandzi,j,top binary variables for the on/off status of each j driver

attached to an i shaft demand in each t operatingperiod

REFERENCESAguilar, O., Perry, S., Kim, J. and Smith, R., 2007, Design and optim-ization of flexible utility systems subject to variable conditions—

Part 1: Modelling framework, Chem Eng Res Des, accepted forpublication.

Brooke, A., Kendrik, D., Meeraus, A. and Raman, R., 1998, GAMS—A User’s Guide, (GAMS Development Corporation, USA).

Del Nogal, F., Townsend, D.W. and Perry, S.J., 2003, Synthesis ofpower systems for LNG plants, GPA Eurpoe Spring Meeting, Bour-memouth, UK.

Maia, L.O.A. and Qassim, R.Y., 1997, Synthesis of utility systemswith variable demands using simulated annealing, Comput ChemEng, 21(9): 947.

Office of Industrial Technologies (OIT), 1998, United States IndustrialMotor Systems Market Opportunities Assessment, (U.S. Depart-ment of Energy).

Papoulias, S.A. and Grosssman, I.E., 1983, A structural optimizationapproach in process synthesis—I. Utility systems, Comput ChemEng, 7(6): 695.

Swaney, R.E. and Grossmann, I.E., 1985a, An index for operationalflexibility in chemical process design. Part I. Formulation andtheory, AIChE J, 31: 621.

Swaney, R.E. and Grossmann, I.E., 1985b, An index for operationalflexibility in chemical process design. Part 2. Computational algor-ithms, AIChE J, 31: 631.

Varvarezos, D.K., Grossmann, I.E. and Biegler, L.T., 1992, An outerapproximation method for multiperiod design optimization, IndEng Chem Res, 31: 1466.

Varvarezos, D.K., Grossmann, I.E. and Biegler, L.T., 1995, A sensi-tivity based approach for the flexibility analysis and design oflinear process systems, Comp Chem Eng, 19: 1305.

The manuscript was received 20 June 2006 and accepted forpublication after revision 30 March 2007.


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